ID  60871 
FullText URL  
Author 
Seita, Kohei
Department of Mathematics, Graduate School of Natural Science and Technology, Okayama University

Abstract  Let G be a ﬁnite group and let V and W be real Gmodules. We call V and W dimequivalent if for each subgroup H of G, the Hﬁxed point sets of V and W have the same dimension. We call V and W are Smith equivalent if there is a smooth Gaction on a homotopy sphere Σ with exactly two Gﬁxed points, say a and b, such that the tangential Grepresentations at a and b of Σ are respectively isomorphic to V and W . Moreover, We call V and W are dSmith equivalent if they are dimequivalent and Smith equivalent. The diﬀerences of dSmith equivalent real Gmodules make up a subset, called the dSmith set, of the real representation ring RO(G). We call V and W P(G)matched if they are isomorphic whenever the actions are restricted to subgroups with prime power order of G. Let N be a normal subgroup. For a subset F of G, we say that a real Gmodule is Ffree if the Hﬁxed point set of the Gmodule is trivial for all elements H of F. We study the dSmith set by means of the submodule of RO(G) consisting of the diﬀerences of dimequivalent, P(G)matched, {N}free real Gmodules. In particular, we give a rank formula for the submodule in order to see how the dSmith set is large.

Keywords  Real Gmodule
Smith equivalence
representation ring
Oliver group

Note  Mathematics Subject Classiﬁcation. Primary 57S25, Secondary 20C15.

Published Date  202101

Publication Title 
Mathematical Journal of Okayama University

Volume  volume63

Issue  issue1

Publisher  Department of Mathematics, Faculty of Science, Okayama University

Start Page  153

End Page  165

ISSN  00301566

NCID  AA00723502

Content Type 
Journal Article

language 
英語

Copyright Holders  Copyright©2021 by the Editorial Board of Mathematical Journal of Okayama University

File Version  publisher

Refereed 
True

Submission Path  mjou/vol63/iss1/9
